ABSTRACT
We start the discussion by a formal theory of perturbation, which will be further elucidated by straightforward calculations.
2.1.1 The Golden Rule
Assume that H0 is a time-independent Hamiltonian and all solutions of the eigenvalue problem H0|k〉 = Ek|k〉 are known; it is straightforward to construct the solution of the equation of motion
i ∂|ψ(t)〉
∂t = H0|ψ(t)〉 (2.1)
which coincides at t = 0 with the given initial state |ψ(0)〉. Under these circumstances, the state vector
|ψ(t)〉 = e−iH0t/|ψ(0) (2.2) represents the general solution of the equation of motion, Eq. 2.1. Since∑
|k〉〈k| = 1
we obtain
|ψ(t)〉 = ∑ k
e−iH0t/|k〉〈k|ψ(0)〉 = ∑ k
e−iEkt/|k〉〈k|ψ(0)〉 (2.3)
We now calculate transition amplitudes between the relevant unperturbed eigenstates due to the appearance of an external perturbation V at t = 0. We transform the equation of motion in the Schro¨dinger picture
i ∂|ψ(t)〉
∂t = (H0 + V )|ψ(t)〉 (2.4)
to the interaction picture by a time-dependent unitary operator
|ψˆ(t)〉 = exp (iH0t/)|ψ(t)〉 (2.5)
so that in the interaction picture, the equation of motion for the state is
i ∂|ψˆ(t)〉
∂t = Vˆ (t)|ψˆ(t)〉 (2.6)
where the new interaction operator is given by
Vˆ (t) = exp (iH0t/)V exp (−iH0t/) (2.7) Next we introduce a time development operator Tˆ (t) such that
|ψˆ(t)〉 = Tˆ (t)|ψˆ(0)〉 (2.8) The time development operator satisfies the integral equation
Tˆ (t) = 1− i
Vˆ (t)Tˆ (t′)dt′ (2.9)
The relationship between the operator expressions in the two pictures is
Tˆ (t) = exp (iH0t/)T (t) exp (−iH0t/) (2.10) The transition matrix elements of the time development operator between eigen-
states of the unperturbed Hamiltonian are
〈q|Tˆ (t)|k〉 = δ(q − k)− i
〈q|Vˆ (t)Tˆ (t′)|k〉dt′
= δ(q − k)− i
〈q|Vˆ (t′)|k′〉〈k′|Tˆ (t′)|k〉dt′ (2.11)
By Eq. 2.10, and note H0|k〉 = Ek|k〉 and H0|q〉 = Eq|q〉,
〈q|Tˆ (t)|k〉 = δ(q − k)− i
ei(Eq−Ek′ )t ′/〈q|V (t′)|k′〉〈k′|Tˆ (t′)|k〉dt′ (2.12)
The strategy of time-dependent perturbation theory is to assume that V is small and proceed with iterating Eq. 2.9 as a power series in terms of Vˆ
Tˆ (t) = 1− i
Vˆ ′dt′ − 1 2
Vˆ ′′dt′′ + · · · (2.13)
and the transition amplitude from initial state |k〉 to final state |q〉 in the perturbation expansion becomes
〈q|Tˆ (t)|k〉 = δ(q − k)− i
− 1 2
× ∫ t′ 0
ei(Ek′−Ek)t ′′/〈k′|V (t′′)|k〉dt′dt′′ + · · · (2.14)
It is equivalent to obtain the above equation by iteratively replacing the term 〈k′|Tˆ (t′)|k〉 on the right side of Eq. 2.12 by the whole right-side expression of
Eq. 2.12, i.e., ∑ k′
= ∫ t′ 0
+ ∑ k′ =k
ei(Eq−Ek′ )t ′′/〈q|V (t′′)|k′〉〈k′|Tˆ (t′′)|k〉dt′′ (2.15)
By Eq. 2.12
〈k|Tˆ (t′)|k〉 = 1− i
ei(Ek−Ek′′ )t ′′/〈k|V (t′′)|k′′〉〈k′′|Tˆ (t′′)|k〉dt′′ (2.16)
The first term on the right side of the above equation results in the second term on the right side of Eq. 2.14. For k′ = k and by Eq. 2.12, the second term on the right side of Eq. 2.15 becomes
〈k′|Tˆ (t′)|k〉 = − i
ei(Ek′−Ek′′ )t ′′/〈k′|V (t′′)|k′′〉〈k′′|Tˆ (t′′)|k〉dt′′ (2.17)
We use Eq. 2.15 to separate the term k′′ = k from k′′ = k in the above equation, which becomes the third term on the right side of Eq. 2.14.
